B
f
C g //D
(2.8)
is said to be aMilnor square if it is a pull-back square and eitherf org is surjective. It is said to besplit if eitherf orghas a section.
LetF be a functor fromCto abelian groups and let
0 //A //B vv //C //0 (2.9)
be a split-extension inC. We say thatF issplit-exact on (2.9) if 0 //F(A) //F(B) //F(C) //0
is (split-)exact. IfA⊂Cis a subcategory and F is split-exact on all split-exact sequences contained inA, then we say that F is split-exact onA.
Lemma 2.11. Let E:C!Spt be a homology theory and (2.8) be a Milnor square.
Assume that ker(f)is E-excisive. Then E maps (2.8)to a homotopy cartesian square.
Lemma2.12. Let F:C!Abbe a functor,A⊂Cbe a subcategory closed under kernels and (2.8)be a split Milnor square in A. Assume that F is split-exact on A. Then the sequence
0−!F(A)−!F(B)⊕F(C)−!F(D)−!0 is split-exact.
3. Real algebraic geometry and split-exact functors
In this section, we recall several results from real algebraic geometry and prove a theorem on the behavior of weakly split-exact functors with respect to proper semi-algebraic surjections (see Theorem 3.14). Recall that a semi-algebraic set is a priori a subset of Rn which is described as the solution set of a finite number of polynomial equalities and inequalities. A map between semi-algebraic sets is semi-algebraic if its graph is a semi-algebraic set. For general background on semi-algebraic sets, consult [2].
3.1. General results about semi-algebraic sets Let us start with recalling the following two propositions.
Proposition 3.1. (See [2, Proposition 3.1]) The closure of a semi-algebraic set is semi-algebraic.
Proposition 3.2. (See [2, Proposition 2.83]) Let S and T be semi-algebraic sets, S0⊂S and T0⊂T be semi-algebraic subsets and f:S!T be a semi-algebraic map. Then f(S0)and f−1(T0)are semi-algebraic.
Note that a semi-algebraic map does not need to be continuous. Moreover, within the class of continuous maps, there are surjective mapsf:S!T for which the quotient topology induced by S does not agree with the topology on T. An easy example is the projection map from{(0,0)}∪{(t, t−1):t>0}to [0,∞). This motivates the following definition.
Definition 3.3. Let S and T be semi-algebraic sets. A continuous semi-algebraic surjectionf:S!T is said to betopological, if for every semi-algebraic map g:T!Qthe compositiongf is continuous if and only ifg is continuous.
Brumfiel proved the following result, which says that (under certain conditions) semi-algebraic equivalence relations lead to good quotients.
Theorem3.4. (See [5, Theorem 1.4]) Let Sbe a semi-algebraic set and let R⊂S×S be a closed semi-algebraic equivalence relation. If π1:R!S is proper, then there exists a semi-algebraic set T and a topological semi-algebraic surjection f:S!T such that
R={(s1, s2)∈S×S:f(s1) =f(s2)}.
Remark 3.5. Note that the properness assumption in the previous theorem is auto-matically fulfilled ifS is compact. This is the case we are interested in.
Corollary 3.6. Let S,S0 and T be compact semi-algebraic sets and f:T!S and f0:T!S0 be continuous semi-algebraic maps. Then,the topological push-out S∪TS0 car-ries a canonical semi-algebraic structure such that the natural maps σ:S!S∪TS0 and σ0:S0!S∪TS0 are semi-algebraic.
For semi-algebraic sets, there is an intrinsic notion of connectedness, which is given by the following definition.
Definition 3.7. A semi-algebraic setS⊂Rk is said to besemi-algebraically connected if it is not a non-trivial union of semi-algebraic subsets which are both open and closed inS.
One of the first results on connectedness of semi-algebraic sets is the following the-orem.
Theorem 3.8. (See [2, Theorem 5.20]) Every semi-algebraic set S is the disjoint union of a finite number of semi-algebraically connected semi-algebraic sets which are both open and closed in S.
Next we come to aspects of semi-algebraic sets and continuous semi-algebraic maps which differ drastically from the expected results for general continuous maps. In fact, there is a far-reaching generalization of Ehresmann’s theorem about local triviality of submersions. Let us consider the following definition.
Definition 3.9. LetSandT be two semi-algebraic sets andf:S!T be a continuous semi-algebraic function. We say thatf is a semi-algebraically trivial fibration if there exist a semi-algebraic setF and a semi-algebraic homeomorphismθ:T×F!S such that fθis the projection onto T.
A seminal theorem is Hardt’s triviality result, which says that away from a subset of T of smaller dimension, every map f:S!T looks like a semi-algebraically trivial fibration.
Theorem3.10. ([2] or [22,§4]) Let Sand T be two semi-algebraic sets and f:S!T be a continuous semi-algebraic function. Then there exists a closed semi-algebraic subset V⊂T with dimV <dimT, such that f is a semi-algebraically trivial fibration over every semi-algebraic connected component of T\V.
We shall also need the following result about semi-algebraic triangulations.
Theorem 3.11. ([2, Theorem 5.41]) Let S⊂Rk be a compact semi-algebraic set, and let S1, ..., Sq be semi-algebraic subsets. There exists a simplicial complex K and a semi-algebraic homeomorphism h:|K|!S such that each Sj is the union of images of open simplices of K.
Remark 3.12. In the preceding theorem, the case where the subsets Sj are closed is of special interest. Indeed, if the subsetsSj are closed, the theorem implies that the triangulation ofS induces triangulations ofSj for eachj∈{1, ..., q}.
The following proposition is an application of Theorems 3.10and 3.11.
Proposition 3.13. Let T⊂Rm be a compact semi-algebraic subset, S be a semi-algebraic set and f:S!T be a continuous algebraic map. Then there exist a semi-algebraic triangulation of T and a finite sequence of closed subcomplexes
∅=Vr+1⊂Vr⊂Vr−1⊂...⊂V1⊂V0=T
such that the following conditions are satisfied:
(i) for each k∈{0, ..., r} we have dimVk+1<dimVk and the map f|f−1(Vk\Vk+1):f−1(Vk\Vk+1)−!Vk\Vk+1
is a semi-algebraically trivial fibration over every semi-algebraic connected component;
(ii) each simplex in the triangulation lies in some Vk and has at most one face of codimension 1which intersects Vk+1.
Proof. We set n=dimT. By Theorem 3.10, there exists a closed semi-algebraic subsetV1⊂T, with dimV1<n, such thatf is a semi-algebraic trivial fibration over every semi-algebraic connected component ofT\V1. Consider nowf|f−1(V1):f−1(V1)!V1 and proceed as before to findV2⊂V1. By induction, we find a chain
∅⊂Vr⊂Vr−1⊂...⊂V1⊂V0=T such thatVk⊂Vk−1 is a closed semi-algebraic subset and
f|f−1(Vk−1\Vk):f−1(Vk−1\Vk)−!Vk−1\Vk
is a semi-algebraically trivial fibration over every semi-algebraic connected component, for allk∈{1, ..., r}. Using Theorem3.11, we may now choose a semi-algebraic triangula-tion ofT such that the subsets Vk are subcomplexes. Taking a barycentric subdivision, each simplex lies inVk for somek∈{0, ..., r}and has at most one face of codimension 1 which intersects the setVk+1.
3.2. The theorem on split-exact functors and proper maps
The following is our main technical result. It is the key to the proofs of Theorems4.19 and7.6.
Theorem 3.14. Let T be a compact semi-algebraic subset of Rk. Let S be a semi-algebraic set and let f:S!T be a proper continuous semi-algebraic surjection. Then, there exists a semi-algebraic triangulation of T such that for every weakly split-exact contravariant functor F:Pol!Set−and every simplex ∆n in the triangulation,we have
ker(F(∆n)!F(f−1(∆n))) =∗.
Proof. Choose a triangulation of T and a sequence of subcomplexes Vk⊂T as in Proposition 3.13. We shall show that ker(F(∆n)!F(f−1(∆n)))=∗for each simplex in the chosen triangulation. The proof is by induction on the dimension of the simplex. The
statement is clear for zero-dimensional simplices, sincef is surjective. Let ∆n be an n-dimensional simplex in the triangulation. By assumption,f is a semi-algebraically trivial fibration over ∆n\∆n−1 for some face ∆n−1⊂∆n. Hence, there exists a semi-algebraic setK and a semi-algebraic homeomorphism
θ: (∆n\∆n−1)×K−!f−1(∆n\∆n−1)
over ∆n\∆n−1. Consider the inclusion f−1(∆n−1)⊂f−1(∆n). Since f−1(∆n−1) is an absolute neighborhood retract, there exists a compact neighborhood N of f−1(∆n−1) inf−1(∆n) which retracts ontof−1(∆n−1). We claim that the setf(N) contains some standard neighborhoodAof ∆n−1. Indeed, assume thatf(N) does not contain standard neighborhoods. Then there exists a sequence in the complement of f(N) converging to ∆n−1. Lifting this sequence, one can choose a convergent sequence in the complement of N converging to f−1(∆n−1). This contradicts the fact that N is a neighborhood, and hence there exists a standard compact neighborhood A of ∆n−1 in ∆n such that f−1(A)⊂N. Since (∆n\∆n−1)×K∼=f−1(∆n\∆n−1), any retraction of ∆nontoAyields a retraction off−1(∆n) ontof−1(A). We have thatf−1(A)⊂N, and thus we can con-clude thatf−1(∆n−1) is a retract off−1(∆n).
By Corollary3.6, the topological push-out f−1(∆n−1) //
f−1(∆n)
∆n−1 //Z
carries a semi-algebraic structure. Moreover, by weak split-exactness, we have
ker(F(Z)!F(∆n−1)×F(f−1(∆n−1))F(f−1(∆n))) =∗. (3.1) Note thatf−1(∆n\∆n−1)⊂Z by the definition of Z. We claim that the natural map σ:Z!∆n is a semi-algebraic split-surjection. Indeed, identify
f−1(∆n\∆n−1) = (∆n\∆n−1)×K, pickk∈K, and consider
Gk={(d,(d, k))∈∆n×Z:d∈∆n\∆n−1}.
ThenGk is semi-algebraic by Proposition 3.1, and therefore defines a continuous semi-algebraic map%k: ∆n!Z which splitsσ. ThusF(σ):F(∆n)!F(Z) is injective, whence ker(F(∆n)!F(∆n−1)×F(f−1(∆n−1))F(f−1(∆n))) =∗, (3.2) by (3.1). But the kernel ofF(∆n−1)!F(f−1(∆n−1)) is trivial by induction, so the same must be true ofF(∆n)!F(f−1(∆n)), by (3.2).
4. Large semi-algebraic groups and the compact fibration theorem